Just because it is theoretically possible that children from poor, under-privileged backgrounds can break the class barrier and become successful, it does not mean this is at all likely.
Imagine a kid from a working class family. His parents have manual labour or low skill jobs and they're not well-educated. They don't read or own books, they partake in only the most low-consciousness entertainment such as Eastenders and they've never discussed or tried to arouse interest in their child in politics, culture, science etc. The kid goes to a low-performing school where the majority of the kids are in the same situation as him and the teaching and resources aren't brilliant. Of course that kid COULD - and some certainly do - decide to work really hard in school, isolate themselves from a lot of peers and family, develop their own interests in extra-curricular educational persuits and opt for further education despite possible discouragement from parents, who might think university is pointless. They certainly won't have any contacts to held them find the best employment, which largely rules them out of certain careers.
But can't you see that as society currently stand that is never going to be the majority of underprivileged kids? It's not because they're lazy or stupid, they are simply going along with the path designated to them from birth because they never knew any better and were never surrounded by any different.
A kid from a rich family will have intelligent, cultured parents with respectable jobs. These parents are going to introduce their children to things that will advantage them from a young age, be it music lessons, books or general discussions. These children will get to go fee-paying schools, receive a lot of attention from teachers due to smaller classes and be surrounded by children from similarly privileged families. Their friends are therefore going to be a positive influence and it will be completely normal and expected for these children to behave in a way that is the completely contrary to that of the working-class child I talked about. The child will be supported and helped in their education, encouraged and expected to attend further education and the parents are likely to have contacts and inside knowledge which will aid their child in finding a good job.
Do you really see no disparity here? No unfair odds? Obviously most people are somewhere in between these two scenarios but you seem to be implying anybody can succeed.